Group #26132434 — subgroups

Subgroups — #26132434

Order 12032.

358 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20245161	C₂	1	2	yes	maximal, complemented
2	#20243438	C₂	1	1	yes	maximal, characteristic
3	#20243434	C₂	1	1	yes	maximal, characteristic
4	#20241484	C₂	1	2	yes	maximal, complemented
5	C₂×C₄₇×#504	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093960	C₂²	1	2	yes	—
7	#15689161	C₂²	1	2	yes	—
8	#15689107	—	2	1	no	—
9	#15689097	—	2	1	no	—
10	#15688946	—	2	4	no	—
11	C₄₇⋊#338	—	2	1	no	—
12	#15688880	—	2	1	no	—
13	C₂×#131×C₄₇	C₂²	1	1	yes	characteristic
14	C₄₇×#504	—	2	4	no	—
15	C₂×M₃₂×C₄₇	C₄	1	2	yes	complemented
16	C₂×M₃₂×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₄₇	C₄	1	2	yes	complemented
18	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×#10625329	—	2	2	no	—
20	#13093983	—	2	2	no	—
21	#13093960	—	2	2	no	—
22	#13093960	—	2	2	no	—
23	#13093958	—	2	2	no	—
24	#13093956	—	2	2	no	—
25	#13093943	—	4	1	no	—
26	#13093941	—	4	1	no	—
27	#13093939	—	4	2	no	—
28	C₂×#41×C₄₇	D₄	1	1	yes	characteristic
29	#131×C₄₇	—	2	2	no	—
30	#131×C₄₇	—	2	2	no	—
31	C₂×M₁₆×C₄₇	D₄	1	1	yes	characteristic
32	C₂×M₁₆×C₄₇	Q₈	1	2	yes	—
33	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
35	M₃₂×C₄₇	—	2	4	no	—
36	M₃₂×C₄₇	D₄	1	4	yes	—
37	C₂×C₁₆×C₄₇	—	2	2	no	—
38	C₂×C₁₆×C₄₇	—	2	4	no	—
39	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
42	#41×C₄₇	—	2	2	no	—
43	#41×C₄₇	—	2	2	no	—
44	C₂×Q₈×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂×D₄×C₄₇	SD₁₆	1	2	yes	—
46	C₂×D₄×C₄₇	SD₁₆	1	1	yes	characteristic
47	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	M₁₆×C₄₇	—	2	2	no	—
50	M₁₆×C₄₇	—	2	4	no	—
51	C₂×C₈×C₄₇	—	2	2	no	—
52	C₂×C₈×C₄₇	—	2	2	no	—
53	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₈×C₄₇	#41	1	2	yes	—
58	C₂²⋊Dic₄₇	—	4	2	no	—
59	C₄⋊Dic₄₇	—	4	2	no	—
60	C₄×Dic₄₇	—	4	2	no	—
61	#10625329	—	4	2	no	—
62	#10625329	—	4	2	no	—
63	C₂×C₄₇⋊C₈	—	4	2	no	—
64	C₁₆×C₄₇	—	2	1	no	—
65	C₁₆×C₄₇	—	2	1	no	—
66	C₁₆×C₄₇	C₂²⋊C₄	1	4	yes	—
67	C₂³×C₄₇	#106	1	2	yes	—
68	C₂³×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
69	Q₈×C₄₇	—	2	2	no	—
70	D₄×C₄₇	—	2	4	no	—
71	D₄×C₄₇	—	2	2	no	—
72	C₂×C₄×C₄₇	—	2	2	no	—
73	C₂×C₄×C₄₇	—	2	2	no	—
74	C₂×C₄×C₄₇	#106	1	2	yes	—
75	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄×C₄₇	#121	1	2	yes	—
77	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₄₇	#135	1	2	yes	—
79	C₂×Dic₄₇	—	8	2	no	—
80	C₈×C₄₇	—	2	2	no	—
81	C₈×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₈×C₄₇	#117	1	1	yes	characteristic
83	C₈×C₄₇	C₂²⋊Q₈	1	2	yes	—
84	C₄₇⋊C₈	—	8	2	no	—
85	#12832	—	47	1	no	maximal, Sylow
86	C₂²×C₄₇	—	2	2	no	—
87	C₂²×C₄₇	—	4	1	no	—
88	C₂²×C₄₇	—	4	2	no	—
89	C₂²×C₄₇	C₂×#106	1	1	yes	characteristic
90	C₂²×C₄₇	#465	1	2	yes	—
91	C₄×C₄₇	—	2	2	no	—
92	C₄×C₄₇	#380	1	1	yes	characteristic
93	C₄×C₄₇	#427	1	1	yes	characteristic
94	C₄×C₄₇	#465	1	2	yes	—
95	Dic₄₇	—	16	2	no	—
96	C₂×#504	D₄₇	1	1	yes	characteristic
97	#2155	—	47	2	no	—
98	#1813	—	47	1	no	—
99	#1812	—	47	1	no	—
100	#1382	—	47	2	no	—
101	C₂×C₄₇	—	2	2	no	—
102	C₂×C₄₇	—	2	1	no	—
103	C₂×C₄₇	—	4	2	no	—
104	C₂×C₄₇	#1807	1	1	yes	characteristic
105	C₂×#131	C₂×D₄₇	1	1	yes	characteristic
106	#504	—	2	4	no	—
107	C₂×M₃₂	Dic₄₇	1	2	yes	complemented
108	C₂×M₃₂	C₂×D₄₇	1	1	yes	characteristic
109	C₂²×C₁₆	Dic₄₇	1	2	yes	complemented
110	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
111	C₂×#108	—	47	2	no	—
112	#427	—	47	2	no	—
113	#413	—	94	1	no	—
114	#410	—	94	1	no	—
115	#365	—	94	4	no	—
116	#338	—	94	1	no	—
117	H(3,ℤ/4)	—	94	1	no	—
118	C₄₇	#12832	1	1	yes	characteristic, complemented, Sylow
119	C₂×#41	D₁₈₈	1	1	yes	characteristic
120	#131	—	2	2	no	—
121	#131	—	2	2	no	—
122	C₂×M₁₆	—	94	2	no	—
123	C₂×M₁₆	Dic₉₄	1	2	yes	—
124	C₂×M₁₆	D₁₈₈	1	1	yes	characteristic
125	C₂²×C₈	C₂×Dic₄₇	1	2	yes	—
126	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	#117	—	94	2	no	—
128	M₃₂	—	2	4	no	—
129	M₃₂	C₂²⋊D₄₇	1	4	yes	—
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	4	no	—
132	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
133	C₂×C₁₆	C₂×Dic₄₇	1	1	yes	characteristic
134	C₂×C₁₆	C₂×Dic₄₇	1	2	yes	—
135	#108	—	94	2	no	—
136	#108	—	94	2	no	—
137	#107	—	94	2	no	—
138	#106	—	94	2	no	—
139	#101	—	188	1	no	—
140	#100	—	188	1	no	—
141	#99	—	188	2	no	—
142	#41	—	2	2	no	—
143	#41	—	2	2	no	—
144	C₂×Q₈	#10625325	1	1	yes	characteristic
145	C₂×D₄	#10625325	1	2	yes	—
146	C₂×D₄	#10625325	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Dic₄₇	1	2	yes	—
148	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
149	M₁₆	—	2	2	no	—
150	M₁₆	—	2	4	no	—
151	M₁₆	—	188	2	no	—
152	M₁₆	—	188	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	188	2	no	—
156	C₂×C₈	C₄⋊Dic₄₇	1	2	yes	—
157	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
158	C₂×C₈	#10625350	1	2	yes	—
159	C₂×C₈	C₂²×Dic₄₇	1	1	yes	characteristic
160	C₂×C₈	C₂×C₂²⋊D₄₇	1	2	yes	—
161	C₄⋊C₄	—	188	2	no	—
162	C₂²⋊C₄	—	188	2	no	—
163	C₄²	—	188	2	no	—
164	C₁₆	—	2	1	no	—
165	C₁₆	—	2	1	no	—
166	C₁₆	C₂²⋊Dic₄₇	1	4	yes	—
167	C₂³	#13093956	1	2	yes	—
168	C₂³	C₂×#10625325	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	376	2	no	—
175	C₂×C₄	#13093956	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
177	C₂×C₄	#13093998	1	2	yes	—
178	C₂×C₄	C₂×#10625325	1	1	yes	characteristic
179	C₂×C₄	#13094046	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	376	2	no	—
182	C₈	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
183	C₈	#13093983	1	1	yes	characteristic
184	C₈	#13094003	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	—	4	1	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂×#13093956	1	1	yes	characteristic
189	C₂²	#15689352	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	752	2	no	—
192	C₄	#15688983	1	1	yes	characteristic
193	C₄	#15689161	1	1	yes	characteristic
194	C₄	#15689352	1	2	yes	—
195	C₂	—	2	2	no	—
196	C₂	—	2	1	no	—
197	C₂	—	4	2	no	—
198	C₂	#20243413	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245161

C₂

yes

maximal, complemented

#20243438

C₂

yes

maximal, characteristic

#20243434

C₂

yes

maximal, characteristic

#20241484

C₂

yes

maximal, complemented

C₂×C₄₇×#504

C₂

yes

maximal, characteristic

C₂×#13093960

C₂²

yes

—

#15689161

C₂²

yes

—

#15689107

—

#15689097

—

#15688946

—

C₄₇⋊#338

—

#15688880

—

C₂×#131×C₄₇

C₂²

yes

characteristic

C₄₇×#504

—

C₂×M₃₂×C₄₇

C₄

yes

complemented

C₂×M₃₂×C₄₇

C₂²

yes

characteristic

C₂²×C₁₆×C₄₇

C₄

yes

complemented

C₂²×C₁₆×C₄₇

C₂²

yes

—

—

—

—

—

—

—

—

—

C₂×#41×C₄₇

D₄

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

D₄

yes

characteristic

C₂×M₁₆×C₄₇

Q₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

D₄

yes

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

SD₁₆

yes

characteristic

C₂×D₄×C₄₇

SD₁₆

yes

—

C₂×D₄×C₄₇

SD₁₆

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

C₂²⋊C₄

yes

—

C₂³×C₄₇

#106

yes

—

C₂³×C₄₇

C₂×SD₁₆

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#106

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₂²⋊Q₈

yes

—

C₄₇⋊C₈

—

#12832

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#106

yes

characteristic

C₂²×C₄₇

#465

yes

—

C₄×C₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

Dic₄₇

—

C₂×#504

D₄₇

yes

—

—

—

100

#1382

—

101

C₂×C₄₇

—

102

C₂×C₄₇

—

103

C₂×C₄₇

—

104

C₂×C₄₇

#1807

yes

characteristic

105

C₂×#131

C₂×D₄₇

yes

characteristic

106

#504

—

107

C₂×M₃₂

Dic₄₇

yes

complemented

108

C₂×M₃₂

C₂×D₄₇

yes

characteristic

109

C₂²×C₁₆

Dic₄₇

yes

complemented

110

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

111

C₂×#108

—

112

#427

—

113

#413

—

114

#410

—

115

#365

—

116

#338

—

117

H(3,ℤ/4)

—

118

C₄₇

#12832

yes

characteristic, complemented, Sylow

119

C₂×#41

D₁₈₈

yes

characteristic

120

#131

—

121

#131

—

122

C₂×M₁₆

—

123

C₂×M₁₆

Dic₉₄

yes

—

124

C₂×M₁₆

D₁₈₈

yes

characteristic

125

C₂²×C₈

C₂×Dic₄₇

yes

—

126

C₂²×C₈

C₂²×D₄₇

yes

characteristic

127

#117

—

128

M₃₂

—

129

M₃₂

C₂²⋊D₄₇

yes

—

130

C₂×C₁₆

—

131

C₂×C₁₆

—

132

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

133

C₂×C₁₆

C₂×Dic₄₇

yes

characteristic

134

C₂×C₁₆

C₂×Dic₄₇

yes

—

135

#108

—

136

#108

—

137

#107

—

138

#106

—

139

#101

—

188

—

140

#100

—

188

—

141

#99

—

188

—

142

#41

—

143

#41

—

144

C₂×Q₈

#10625325

yes

characteristic

145

C₂×D₄

#10625325

yes

—

146

C₂×D₄

#10625325

yes

characteristic

147

C₂²×C₄

C₄⋊Dic₄₇

yes

—

148

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

149

M₁₆

—

150

M₁₆

—

151

M₁₆

—

188

—

152

M₁₆

—

188

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

—

188

—

156

C₂×C₈

C₄⋊Dic₄₇

yes

—

157

C₂×C₈

C₂×Dic₉₄

yes

characteristic

158

C₂×C₈

#10625350

yes

—

159

C₂×C₈

C₂²×Dic₄₇

yes

characteristic

160

C₂×C₈

C₂×C₂²⋊D₄₇

yes

—

161

C₄⋊C₄

—

188

—

162

C₂²⋊C₄

—

188

—

163

C₄²

—

188

—

164

C₁₆

—

165

C₁₆

—

166

C₁₆

C₂²⋊Dic₄₇

yes

—

167

C₂³

#13093956

yes

—

168

C₂³

C₂×#10625325

yes

characteristic

169

Q₈

—

170

D₄

—

171

D₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

376

—

175

C₂×C₄

#13093956

yes

—

176

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

177

C₂×C₄

#13093998

yes

—

178

C₂×C₄

C₂×#10625325

yes

characteristic

179

C₂×C₄

#13094046

yes

—

180

C₈

—

181

C₈

—

376

—

182

C₈

C₂×C₂²⋊Dic₄₇

yes

characteristic

183

C₈

#13093983

yes

characteristic

184

C₈

#13094003

yes

—

185

C₂²

—

186

C₂²

—

187

C₂²

—

188

C₂²

C₂×#13093956

yes

characteristic

189

C₂²

#15689352

yes

—

190

C₄

—

191

C₄

—

752

—

192

C₄

#15688983

yes

characteristic

193

C₄

#15689161

yes

characteristic

194

C₄

#15689352

yes

—

195

C₂

—

196

C₂

—

197

C₂

—

198

C₂

#20243413

yes

characteristic

	subgroup	G/H
center	C₄	#15689161
commutator	C₂×C₈×C₄₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₂×C₄₇×#504	C₂
socle	C₂×C₄₇	#1807
prime radical	C₂×#41×C₄₇	D₄
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₂×#504	D₄₇
hypocenter	C₄₇	#12832
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

subgroup

G/H

center

C₄

#15689161

commutator

C₂×C₈×C₄₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₂×C₄₇×#504

C₂

socle

C₂×C₄₇

#1807

prime radical

C₂×#41×C₄₇

D₄

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ C₂×#504

N	H ≅ G/N	type
1. #20245161	C₂	semidirect
4. #20241484	C₂	semidirect
15. C₂×M₃₂×C₄₇	C₄	semidirect
17. C₂²×C₁₆×C₄₇	C₄	semidirect
107. C₂×M₃₂	Dic₄₇	semidirect
109. C₂²×C₁₆	Dic₄₇	semidirect
118. C₄₇	#12832	semidirect

H ≅ G/N

type

1. #20245161

C₂

semidirect

4. #20241484

C₂

semidirect

15. C₂×M₃₂×C₄₇

C₄

semidirect

17. C₂²×C₁₆×C₄₇

C₄

semidirect

107. C₂×M₃₂

Dic₄₇

semidirect

109. C₂²×C₁₆

Dic₄₇

semidirect

118. C₄₇

#12832

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20245161	2
#20243438	2
#20243434	2
#20241484	2
C₂×C₄₇×#504	2
#12832	47

subgroup

index

C₂×C₄₇×#504

Subgroups — #26132434

Conjugacy classes of subgroups

Complements

Maximal subgroups